Improved Lower Bounds for Ginzburg-Landau Energies via Mass Displacement

We prove some improved estimates for the Ginzburg-Landau energy (with or without magnetic field) in two dimensions, relating the asymptotic energy of an arbitrary configuration to its vortices and their degrees, with possibly unbounded numbers of vortices. The method is based on a localisation of the ``ball construction method" combined with a mass displacement idea which allows to compensate for negative errors in the ball construction estimates by energy ``displaced" from close by. Under good conditions, our main estimate allows to get a lower bound on the energy which includes a finite order ``renormalized energy" of vortex interaction, up to the best possible precision i.e. with only a $o(1)$ error per vortex, and is complemented by local compactness results on the vortices. This is used crucially in a forthcoming paper relating minimizers of the Ginzburg-Landau energy with the Abrikosov lattice. It can also serve to provide lower bounds for weighted Ginzburg-Landau energies.